Poisson-Lie groups, bi-Hamiltonian systems and integrable deformations

Abstract

Given a LiePoisson completely integrable bi-Hamiltonian system on ℝ, we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial LiePoisson system on a non-abelian PoissonLie group G of dimension n, where n ∈ ℝ is the deformation parameter. Moreover, we show that from the two multiplicative (PoissonLie) Hamiltonian structures on G- that underly the dynamics of the deformed system and by making use of the group law on G, one may obtain two completely integrable Hamiltonian systems on G×G By construction, both systems admit reduction, via the multiplication in G, to the deformed bi-Hamiltonian system in G. The previous approach is applied to two relevant LiePoisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.